Slow focus sensor for the Keck I laser guide star adaptive optics system using focal plane wavefront sensing

Abstract: Laser guide stars (LGSs) have been deployed for the last 20-30 years in ground-based astronomical telescopes to overcome the limited sky coverage of classical adaptive optics (AO) systems. Unfortunately, slow altitude drifts of the sodium layer compromise focus measurements, generating the so-called slow focus error, and, consequently, a natural guide star (NGS) is needed to compensate for that error. Our goal is to develop and operationalize a focal plane wavefront sensing (FPWFS) technique for slow focus tracking for the Keck I telescope, which can significantly increase sky coverage and allow slow focus tracking at higher frequencies, reducing the lag error. We develop, characterize, and compare three different FPWFS algorithms, namely Gerchberg-Saxton (GS), linearized focal plane technique (LiFT), and Gaussian fit (Gf). These algorithms were studied for the specific purpose of slow focus sensing in the NIR (H and K bands) using numerical simulations and data collected at Keck in 2025 (bench and on-sky). The three algorithms were studied and characterized against different criteria such as linearity, computational costs, and resistance to low signal-to-noise ratio and/or residuals. From the results obtained, the main candidate for an on-sky deployment was GS. On-sky tests showed promising results, with GS successfully compensating for purposely introduced focus errors, even under the presence of high turbulence conditions. This work can also be extrapolated to other existing 8-10 m class telescopes, or even future 30-40 m class telescopes, where the use of FPWFS can significantly improve sky coverage and reduce the lag error.

• The paper demonstrates a Gerchberg–Saxton focal-plane wavefront sensor that uses existing TRICK images and dichroic-induced astigmatism to track slow focus without hardware changes.
• Bench tests showed Gerchberg–Saxton achieved 33 nm rms focus error under simulated high-order turbulence, outperforming Gaussian-fit and LiFT estimators at 66 and 102 nm rms, respectively.
• On-sky closed-loop tests at 1 Hz compensated injected focus offsets, achieving a post-convergence mean of −22 nm and 71 nm rms over approximately 10 minutes, while simulations indicate faster cadence could substantially reduce sodium-layer lag error.

The slow focus error arising from altitude drifts of the mesospheric sodium layer is a persistent constraint in laser guide star (LGS) adaptive optics (AO): because the LGS is at finite distance, its focus does not track the science target, and an additional natural guide star (NGS) must supply low-bandwidth focus measurements. This paper by Salgueiro et al. (2602.15746) develops, characterizes, and demonstrates on-sky a replacement for the Shack-Hartmann-based low-bandwidth wavefront sensor (LBWFS) currently used for slow focus tracking at Keck I, using focal plane wavefront sensing (FPWFS) on images from the existing near-infrared tip-tilt sensor, TRICK. No hardware modifications are required: TRICK's dichroic transmission introduces approximately 200 nm rms of astigmatism (0°) that naturally serves as phase diversity, resolving the sign ambiguity of even modes such as focus.

Motivation and expected gains

The current LBWFS is a 20×20 Shack-Hartmann WFS (with a 5×5 mode on fainter stars) receiving only 10% of the visible light. Because FPWFS is a full-pupil technique, the authors estimate a flux gain of roughly 3000 relative to a single Shack-Hartmann sub-aperture when all losses are accounted for — corresponding to a magnitude gain of $\Delta m \approx -8.7$ in the H band. The paper is careful to note this is a rough estimate that neglects relative stellar flux across bands and the higher signal-to-noise ratio (S/N) required by FPWFS; moreover, the authors concede the magnitude gain may not be fully realized since tracking the tip-tilt star itself could become limiting.

Beyond sky coverage, FPWFS enables faster correction cadence. Using measured sodium layer statistics (PSD with $\alpha = 35\,\mathrm{m^2 Hz^{-1}}$, $\beta = -1.9$), the authors show that correcting every 35 s yields ~50 nm rms lag error, reducible to ~18 nm rms at 5 s cadence. Since the lag error scales as $D^2$, this cadence argument becomes more stringent for 30–40 m telescopes, motivating applicability to extremely large telescopes (ELTs).

Algorithms under test

Three single-image phase-retrieval algorithms were compared:

• Gerchberg-Saxton (GS): iterative pupil/focal-plane propagation with amplitude constraints, equivalent to Fienup's error-reduction algorithm. The initial phase estimate comes from a diversity calibration using in-focus images, which accounts for non-common path aberrations (NCPAs) following the approach demonstrated at VLT/MUSE-NFM (Kuznetsov et al., 2024).
• Linearized focal plane technique (LiFT): maximum-likelihood estimation under Gaussian noise using a small-phase approximation, estimating four modes (tip-tilt, focus, astigmatisms).
• Gaussian fit (Gf): a simplified estimator exploiting the fact that an astigmatic PSF stretches along perpendicular axes according to focus sign, mapping the ratio $\sigma_y/\sigma_x$ to the focus coefficient via a calibrated gain.

A notable methodological finding concerns sampling. TRICK operates below 0.5 Nyquist sampling (50 mas pixels in both H and K bands), while GS conventionally requires ≥1 Nyquist. The authors apply GS with a deliberately mismatched Nyquist-sampled model and find linearity preserved, requiring only a stable gain correction (~4.5–5.5). The same behavior holds for LiFT. The paper explicitly states that no theoretical explanation for this robustness has been found; it remains an empirical result justified heuristically by the observation that low-order modes manifest near the PSF core, whose morphology is largely insensitive to sampling errors. The required gain grows linearly (~slope 1.5–2) with the committed sampling error factor.

Simulation and bench results

Under ideal conditions, all three algorithms are linear within ±200 nm rms of focus, both with 8×8-pixel and reduced 4×4-pixel fields of view, and each supports several focus estimates per second on a desktop computer (using 16×16-pupil models as the accuracy/speed trade-off). Under photon and readout noise, simulated H-band limiting magnitudes at a 50 nm rms threshold are m = 18 (GS), 19.5 (Gf), and 20 (LiFT). For a representative m ≈ 15.5–16 star, replacing the LBWFS's 30 s integration with a 1 s FPWFS integration would reduce lag error by ~37 nm rms.

Bench tests at Keck confirmed these trends but with degraded linearity relative to simulations. The decisive discriminator emerged in open-loop slow-focus tracking with injected high-order residuals (~200–300 nm rms, generated from DM command sequences reproducing median Keck turbulence): GS achieved a 33 nm rms error with regression coefficient $R^2 = 0.74$, whereas Gf and LiFT degraded to 66 nm rms ($R^2 = 0.40$) and 102 nm rms ($R^2 = 0.22$), respectively. High-order residuals therefore compromise the stability and linearity of Gf and LiFT substantially more than GS, which motivated the selection of GS for closed-loop deployment. The authors note that incorporating residual power spectral densities into LiFT or GS was not investigated, since regularization risks biasing the solution, though they flag it as a possible avenue.

Two further caveats bear on the LiFT comparison. First, the favorable on-sky focus-ramp result recently obtained with LiFT at the VLT was achieved under more-than-twice-Nyquist sampling, unlike the sub-0.5 Nyquist TRICK configuration; LiFT should not be excluded under better-sampled conditions. Second, bench tests indicated that estimating more LiFT modes (8–10) can degrade performance due to model over-fitting and inter-mode cross-talk under undersampling, making the optimal mode count operationally ambiguous.

On-sky demonstration

Closed-loop on-sky tests were conducted on February 8, 2025, during the last hour of the half-night with a bright tip-tilt star ($m_V = 10$) and poor seeing (~1″ in K band). The seeing made centering difficult and limited usable data to roughly 10 minutes — a small dataset, which the results should be weighed against. The loop ran at 1 Hz using 1 s numerically integrated exposures from 1 ms short exposures. After an initial ~100 s convergence period, intentionally introduced defocus offsets via the LGS WFS focus stage were correctly identified and compensated by GS, with retrieved focus values remaining mainly within ±100 nm rms and not diverging despite severe turbulence. The mean and standard deviation after convergence were −22 nm and 71 nm rms respectively, the negative mean reflecting that all injected offsets were applied in one direction. Loop gain was progressively increased as stability was verified.

Limitations and open questions

Several limitations qualify the conclusions. The empirical nature of the sampling-error tolerance in GS lacks theoretical grounding, leaving the question of why a Nyquist-mismatched model preserves focus linearity unanswered. The on-sky validation covers only ~10 minutes with a bright star under one night's conditions; no direct on-sky comparison against the LBWFS has yet been performed, so claims about improved sky coverage and lag reduction remain supported primarily by simulation and flux-budget estimates. The diversity calibration depends on NCPA stability over time; the authors propose re-calibration every ~10 minutes on-sky but have not yet demonstrated this operationally. Finally, whether Gf or LiFT would outperform GS under different operational configurations — particularly better-sampled or lower-residual conditions — remains open.

Conclusion

This work establishes GS-based FPWFS on the TRICK sensor as a viable, hardware-free slow focus sensor for the Keck I LGS-AO system, demonstrating closed-loop focus compensation on-sky under challenging seeing. The key quantitative finding is GS's markedly superior robustness to high-order residuals compared to LiFT and Gf, together with empirically demonstrated tolerance to sub-0.5 Nyquist sampling. Full operationalization awaits further on-sky validation, including head-to-head comparison with the LBWFS, and the approach offers a transferable proof of concept for other 8–10 m facilities and ELT-scale systems where the $D^2$ scaling of sodium-layer-induced focus error makes rapid, sensitive slow-focus sensing increasingly critical.